arXiv subjects

Maximiliano Camporino

Publications and source records attributed to Maximiliano Camporino.

2 recordsLinked to original sources

arXiv2015

Lifting Galois representations to ramified coefficient fields

Let $p>5$ be a prime integer and $K/\mathbb{Q}_p$ a finite ramified extension with ring of integers $\mathcal{O}$ and uniformizer $π$. Let $n>1$ be a positive integer and $ρ_n:G_\mathbb{Q} \to \text{GL}_2(\mathcal{O}/π^n)$ be a continuous Galois representation. In this article we prove that under some technical hypotheses the representation $ρ_n$ can be lifted to a representation $ρ:G_\mathbb{Q} \to \text{GL}_2(\mathcal{O})$. Furthermore, we can pick the lift restriction to inertia at any finite set of primes (at the cost of allowing some extra ramification) and get a deformation problem whose universal ring is isomorphic to $W(\mathbb{F})[[X]]$. The lifts constructed are "nearly ordinary" (not necessarily Hodge-Tate) but we can prove the existence of ordinary modular points (up to twist).

math.NT↗

arXiv2014

Congruences between modular forms modulo prime powers

Given a prime $p \ge 5$ and an abstract odd representation $ρ_n$ with coefficients modulo $p^n$ (for some $n \ge 1$) and big image, we prove the existence of a lift of $ρ_n$ to characteristic $0$ whenever local lifts exist (under some technical conditions). Moreover, we can chose the inertial type of our lift at all primes but finitely many (where the lift is of Steinberg type). We apply this result to the realm of modular forms, proving a level lowering theorem modulo prime powers and providing examples of level raising. In particular, our method shows that given a modular eigenform $f$ without Complex Multiplication or inner twists, for all primes $p$ but finitely many, and for all positive integers $n$, there exists another eigenform $g\neq f$, which is congruent to $f$ modulo $p^n$.

math.NT↗